Finite type morphisms with discrete fibers are quasi-finite.
Lemma 29.20.8. Let f : X \to S be a morphism of schemes. Assume f is locally of finite type. Then the following are equivalent
f is locally quasi-finite,
for every s \in S the fibre X_ s is a discrete topological space, and
for every morphism \mathop{\mathrm{Spec}}(k) \to S where k is a field the base change X_ k has an underlying discrete topological space.
Proof. It is immediate that (3) implies (2). Lemma 29.20.6 shows that (2) is equivalent to (1). Assume (2) and let \mathop{\mathrm{Spec}}(k) \to S be as in (3). Denote s \in S the image of \mathop{\mathrm{Spec}}(k) \to S. Then X_ k is the base change of X_ s via \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(\kappa (s)). Hence every point of X_ k is closed by Lemma 29.20.4. As X_ k \to \mathop{\mathrm{Spec}}(k) is locally of finite type (by Lemma 29.15.4), we may apply Lemma 29.20.6 to conclude that every point of X_ k is isolated, i.e., X_ k has a discrete underlying topological space. \square
Comments (1)
Comment #3032 by Brian Lawrence on